CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 60 (1986) 243-251 243 North-Holland ON THE NUMBER OF TRIANGLES IN SIMPLE ARRANGEMENTS OF PSEUDOLINES IN THE REAL PROJECTIVE PLANE Jean-Pierre ROUDNEFF Universite’ Pierre et Marie Curie, U. E. R. 48, Matht%natiques (Combinatoire), 75005 Paris, France Received 4 September 1984 For any simple arrangement Se of n 2 4 pseudolines in the real projective plane, it is easy to show that the number P3 of triangles in the 2-dimensional cell complex associated with & satisfies P3 S $n(n - 1). In this paper we construct infinite classes of simple arrangements for which P3 = fn(n - l), answering negatively a conjecture of Griinbaum. An interpretation of our results in terms of oriented matroids is also given. 1. Introduction In the real projective plane P, an arrangement J& of pseudolines as defined in [3] is a collection of n 2 3 simple closed curves such that every two curves have exactly one point in common at which they cross. This notion is a direct generalization of the well-known arrangements of straight lines. An arrangement ~4 is called simple if no point of P belongs to more than two pseudolines. A simple arrangement induces a decomposition of P into a 2- dimensional cell complex %. The number of faces with k vertices in % will be denoted by Pk. We call triangles the faces with 3 vertices. In a simple arrangement of at least 4 pseudolines, two triangles cannot have a common edge. It follows that: (i) P3 G $z(n - 1) and we have: (ii) P3 = $z(n - 1) if and only if every edge is adjacent to exactly one triangle. For instance the unique simple arrangement of 4 pseudolines satisfies (ii): we will say it is a tight simple arrangement. As easily seen, (ii) cannot hold for n 3 2 (mod 3) nor for odd n, so that (i) is not in general the best bound for a given n. Our purpose in the present paper is to construct infinite classes of simple arrangements satisfying (ii). As a consequence (i) cannot be improved for all n 2 4 by replacing $n(n - 1) by another polynomial. In [3, p. 27 and S-561 Grtinbaum proposed the following conjecture. Conjecture. If & is a simple arrangement with n(a) = n > 6, then [$n(n - l)] for n = 4 (mod 6), P,(~)s([$z(n-2)] forn$4(mod6). 0012-365X/86/$3.50 @ 1986, Elsevier Science Publishers B.V. (North-Holland) 244 J.-P. Roudneff We will answer negatively to this conjecture, even if we restrict ourselves to arrangements of straight lines. The constructions we are going to expose are recursive and allow us to complete Griinbaum’s table [3, p. 561. 2. Constructions of simple arrangements In this section we assume that & is a simple arrangement of II > 4 pseudolines satisfying (ii). Lemma 2.1. Zf n > 4, then P4 = 0. Proof. If we suppose that there exists a face with 4 vertices in %, then we have the configuration shown in Fig. 1. The points A and A’ must be equal and we must have n = 4. 0 \ : \ A A' 1 I 0I I Fig. 1. Theorem 2.1. Given a simple arrangement of n > 4 pseudolines satisfying (ii) one can construct a simple arrangement of 2(n - 1) pseudolines with the same property. Proof. Let d be a simple arrangement of n > 4 pseudolines satisfying (ii). We construct a simple arrangement of 2(n - 1) pseudolines with the same property. Using Theorem 3.3 of [3] we may assume that each pseudoline of J& is polygonal and each face is a convex polygon. Let D, be an arbitrarily chosen pseudoline of &: D, will be considered as the “pseudoline at infinity”. The idea is to replace each pseudoline D of S& D # D, by two pseudolines D1, D2 ‘parallel’ to D (i.e., D1 and Dz crossing each other at D rl D,). More precisely let F be a face of ‘% with k vertices: by Lemma 2.1, kf 4. If F is not bounded by D,, we draw a star polygon in F by joining vi to v3, . , uk-_2 to vk, v&l to VI, vk to v2 where vi, . , vk denote the middles of the edges of F (Fig. 2). If F is bounded by D,, let [a, b] be its edge contained in D, and vi, . , vk_l be the middles of its other edges as shown on Fig. 3. We join b to v2 and vi to a On the number of triangles in simple arrangements 245 “1 r-- \ 7, “3’A “2 \ I “4 k=3 k=5 k=6 Fig. 2. b b % a b % a “3 k=3 k=5 k=6 Fig. 3. if k = 3 and b to u2, . , vk-3 to v&l, u&2 to a if k # 3. When these operations have been done for each face, we remove the elements of ti (except the Vi) and the remaining configuration A?’ is a regular Eulerian graph of degree 4 imbedded in P. Clearly &’ can be considered as an arrangement of m 2 1 closed curves with a crossing at each vertex. Now to each pseudoline D # D, of .& we associate the two simple curves D,, D2 of &’ crossing each other at D fl D, (Fig. 4). Conversely every curve D ’ of LZ~’can be considered as one of the two simple curves D1, D2 associated with some pseudoline D # D, of d’, so that m = 2(n - 1). With this interpretation it is clear that every two distinct curves of d’ cross at least once, and in fact exactly once as it can be shown by a topological argument or by a counting argument consisting in showing that the number of ‘r; / ’ \ / I \ I. D1 __ yL -______,c__ D \ / \ / : \ / , \ / \ 1 \ / ‘\ 1 \ I \ / i ‘A >, 1’‘\ I ‘\ 1 \ I I Fig. 4. 246 J.-P. Roudneff vertices of &’ is $m(m - 1). Hence, the configuration ~4’ can be considered as a simple arrangement of 2(n - 1) pseudolines. Moreover each edge of d’ is adjacent to exactly one triangle provided P4 = 0 and this is the case when IZ> 4, so that &’ satisfies (ii). 0 Remark 2.1. The construction also holds for the unique simple arrangement of 4 pseudolines. Example 2.1. We apply the construction of Theorem 2.1 to the unique tight simple arrangement of 6 pseudolines. We obtain the tight simple arrangement of 10 pseudolines shown on Fig. 5. Repeating the construction, we obtain the stretchable arrangement of 18 pseudolines shown on Fig. 6. Another construction is available which allows us to prove the following result. Theorem 2.2. Given a simple arrangement of n > 4 pseudolines satisfying (ii) one can construct a simple arrangement of (3n - 2) pseudolines with the same property. Proof. Let d be a simple arrangement of n > 4 pseudolines satisfying (ii). We construct a simple arrangement of (3n - 2) pseudolines with the same property. We use the notations of Theorem 2.1. If D, has been chosen, the idea is to associate with every pseudoline D of ~4, D # D, two pseudolines D1, D2 ‘almost parallel’ to D. More precisely we decompose a face F of % in the following graphical way. Fig. 5. A tight simple arrangement of 10 pseudolines. On the number of triangles in simple arrangements 241 Fig. 6. A tight simple arrangement of 18 straight lines. If F is not bounded by D,, see Fig. 7. In the remaining cases, we fix a pseudo-hemisphere H defined by D, and for each face bounded by D, we consider in fact the representative F included in H in order to decompose it in a non-symmetrical way (if k # 3) (Fig. 8): DC0 D, DC0 k=3 k=5 k=6 Fig. 8. 248 J.-P. Roudneff When these operations have been done for each face we have created 2(n - 1) new pseudolines and it is not hard to show they form with ti a simple arrangement .G?’of (3n - 2) pseudolines. &’ satisfies (ii) provided P4 = 0 and this condition holds if n > 4. 0 Remark 2.2. The construction is also available for n = 4. Example 2.2. By applying the construction of Theorem 2.2 to the tight simple arrangement of 6 pseudolines, we obtain the configuration of 16 pseudolines discovered by Simmons [6] in 1971 and shown on Fig. 9. Fig. 9. A tight simple arrangement of 16 pseudolines (incuding the pseudoline at infinity). 3. Applications By repeatedly applying the construction described in Theorem 2.1 starting from the simple arrangement of 4 pseudolines, we generate tight simple arrangements of n = 4, 6, 10, 18, 34. pseudolines and more generally of n = 2k + 2 pseudo- lines. We can state: Theorem 3.1. The equality P3 = $n(n - 1) is achieved for an infinite number of values of n. Hence P3 s $(n - 1) is the best possible polynomial bound. On the number of triangles in simple arrangements 249 In [3], Grtinbaum proposed the following conjecture. Conjecture. Zf d isa simple arrangement with n(a) = n > 6, then [$z(rz - l)] for n = 4 (mod 6) P3(d)S[[$z(n-2)] fornf4(mod6)’ The above results show that this conjecture is false for n = 18 even if we restrict ourselves to simple arrangements of straight lines as shown on Fig. 6. Theorem 3.2. A simple arrangement of n > 5 pseudolines satisfying P3 < $(n - 1) also satisfies P3 G [+(n’ - n - 5)].